Question

How are self-replicating molecules, such as those in the RNA world hypothesis, essential to how evolutionists believe life began on Earth

Answer 1

It's not clear they are. They would be very useful, in the sense that it becomes much easier to understand how life can arise from nonlife (abiogenesis) if you have an example of a molecule that can catalyze its own duplication. This is why the discovery of ribozymes was so exciting -- well, one reason.

However, it is also certainly possible that the first self-replicating systems were systems of molecules. You, after all, are simply a system of self-replicating molecules. The trick is that proteins help nucleic acids (RNA and DNA) to replicate, and then the nucleic acids help the proteins to duplicate. It's actually much easier to imagine a small collection of molecules that self-replicate than just one. The problem is in then understanding how the system got together in the first place -- why would molecules that can potentially self-replicate be found in close proximity? We can imagine chemical conditions under which proteins form spontaneously, or under which nucleic acids form spontaneously, but not really both at the same time. Did one get transported to the other, somehow? It seems odd.

Abiogenesis remains the hardest problem in geobiology, and I don't think there has been much progress since the 1950s. The leading explanations remain: (1) some very, very unlikely weird chance chemical event or two, (2) panspermia (life came here from another planet, or solary system, aboard a meteorite or comet), and of course (3) a miracle.

The discovery of life on another planet would be of profound importance in helping to resolve the situation.